An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

Three-Component Reaction of Triphenylphosphine,Dimethyl Acetylenedicarboxylate,and Aldehyde Benzoylhydrazones: An Efficient One-Pot Synthesis of Stable Phosphorus Ylides

Protonation of the reactive 1:1 intermediate produced in the reaction between dimethyl acetylenedicarboxylate and triphenylphosphine by benzoylhydrazones leads to vinylphosphonium salts, which undergo Michael addition with the conjugate base of the NH acid to produce highly fanctionalized, salt-free phosphorus ylides in excellent yields.     

An Efficient,One-Pot Synthesis of Dialkyl 5-Hydroxy-4-aryl-2,5-dihydrofuran-2,3-dicarboxylate Derivatives

The reaction between arylglyoxalmonohydrates, dialkyl acetylenedicarboxylates, and triphenylphosphine is described as a simple and efficient method for the synthesis of 5-hydroxy-4-aryl-2,5-dihydrofuran-2,3-dicarboxylate derivatives.     

Dynamic and rheological properties of soft biological cell suspensions

Quantifying dynamic and rheological properties of suspensions of soft biological particles such as vesicles, capsules, and red blood cells (RBCs) is fundamentally important in computational biology and biomedical engineering. In this review, recent studies on dynamic and rheological behavior of soft biological cell suspensions by computer simulations are presented, considering both unbounded and confined shear flow. Furthermore, the hemodynamic and hemorheological characteristics of RBCs in diseases such as malaria and sickle cell anemia are highlighted.